Fractal-based techniques have opened new avenues in the analysis of geophysical data. On the other hand, there is often a lack of appreciation of both the statistical uncertainty in the results, and the theoretical properties of the stochastic concepts associated with these techniques. Several examples are presented which illustrate suspect results of fractal techniques. It is proposed that concepts used in fractal analyses are stochastic concepts and the fractal techniques can readily be incorporated into the theory of stochastic processes. This would be beneficial in studying biases and uncertainties of results in a theoretically consistent framework, and in avoiding unfounded conclusions. In this respect, a general methodology for theoretically justified stochastic processes, which evolve in continuous time and stem from maximum entropy production considerations, is proposed. Some important modelling issues are discussed with focus on model identification and fitting, which are often made using inappropriate methods. The theoretical framework is applied to several processes, including turbulent velocities measured every several microseconds and hydroclimatic processes, whose proxy reconstructions can provide information for time scales up to millions of years.